Attractor
Attractor (drag from Latin ad trahere = toward yourself) is a concept from the theory of dynamical systems and describes a subset of the phase space (ie, a certain number of states ) on which a dynamic system moved over the course of time, and the under the dynamics of this system is not left. That is, a set of variables approaches over time ( asymptotically ) a certain value, a curve, or more complex (that is, a region in the n-dimensional space ), and then remains in the further course of time in the vicinity of this attractor. An attractor appears as a clearly recognizable structure. Colloquially you could by a kind of " steady state " speak of a system ( which also periodic, wavelike recurring states or other recognizable patterns can be meant ), ie a state to which moved towards a system. The opposite is called an attractor repellor or negative attractor. Application will find the terms in physics and biology.
Technical term
The set of all points of the phase space who strive for the same attractor of the dynamics, ie of attraction or catchment area of this attractor.
Well-known examples are the Lorenz attractor, Rössler attractor and the zeros of a differentiable function, which are attractors of the corresponding Newton method.
Dynamic systems are often set up as mathematical models of physical or other processes of the real world. Examples are the flow behavior of liquids and gases, movements of celestial bodies under mutual influence of gravity, population sizes of organisms under consideration of the predator-prey relationship or the business performance indicators under the influence of market forces. Dynamic systems are defined by the description of the state change as a function of time t. For the mathematical definition of the real system is often considered a highly simplified form. The reason for that can be described by the global attractor here the long-term behavior of the dynamical system, with physical and technical systems is often dissipation, especially friction.
A distinction between continuous and discrete dynamic systems, depending on whether the state change is defined at fixed time steps () or as a continuous process (). The state is represented by any number of state variables, these are the dimensions of the phase space. Each state so that a point in the phase space, discrete systems are sets of isolated points, continuous systems are represented by curves ( trajectories).
A mixed system of continuous and discrete subsystems - then continuously with - discrete dynamics - is also referred to as a hybrid dynamic system. Examples of such variable structure dynamics can be found in process engineering (eg Dosiervorlagesysteme ). The mathematical description is given by hybrid models, for example by switching differential equations. The trajectories in phase space are i.allg. not continuous ( it show " kinks " and discontinuities in the trajectories).
In the investigation of dynamical systems one is mainly interested in the behavior for at a given initial state. The limit value in this case is referred to as an attractor. Typical and common examples of attractors are:
- Asymptotically stable fixed points: the system approaches increasingly a given final state in which the momentum comes to a halt, it creates a static system. A typical example of such a system is a damped pendulum which is close to the idle state at the lowest point.
- ( asymptotically ) stable limit cycles: The final state is the sequence always the same states that are run periodically (periodic orbits). One example is the simulation of the predator-prey relationship, which amounts for certain parameters of the feedback on a periodic rise and fall in population sizes.
For a hybrid dynamic system with the surface of a chaotic dynamic n- simplex has been identified as the attractor.
- ( asymptotically stable ) Grenztori: If several incommensurate frequencies on each other, so the trajectory is not closed, and the attractor is a Grenztorus which is asymptotically completely filled by the trajectory. The attractor corresponding to this time range is more or less periodically, there is no real period, but the frequency spectrum consists of sharp lines.
These examples are attractors which have an integral dimension in phase space. The existence of attractors with complicated structure was indeed long been known, but they were regarded initially as unstable special cases, the occurrence of which is only observed in certain choice of the initial state and the system parameters. This changed with the definition of a new, special type of attractor:
- Strange Attractor: In its final state, the system often shows a chaotic behavior (there are exceptions, however, such as quasi- periodically driven nonlinear systems ). The strange attractor can not be described in a closed geometric shape and has no integer dimension. Attractors of nonlinear dynamical systems then have a fractal structure. An important feature is the chaotic behavior, that is, even the small change in the initial state leads subsequently to significant state changes. The most prominent example is the Lorenz attractor, which was discovered in the modeling of air currents in the atmosphere.